1. Groupe de Physique des Atomes Refroidis Daniel Hennequin Olivier Houde Optical lattices Philippe Verkerk Laboratoire de Physique des Lasers, Atomes et Molécules Université de Lille 1 ; Villeneuve d’Ascq ; France A lot of work done in the former group of Gilbert Grynberg at ENS.

2. Reactive Force (dipole force) Intensity  =  L -  0 Standing wave  > 0 : « blue » detuning I, U z U : optical potential (light shifts) Optical Lattices

3. Outlook I. Dissipative optical lattices 1D 2D 3D more D II. Non dissipative optical lattices III. Instabilities in a MOT

4. 1D Dissipative Optical Lattice The original one : Sisyphus cooling J=1/2 J=3/2 1/3 1 -+ z E y E x  =  L -  

5. Sisyphus Cooling

6. Quantum Picture  = 2 √ E U vibR0 Pump-Probe spectroscopy Y. Castin & J. Dalibard EuroPhys. Lett. 14, 761 (1991)

7. Two-photon transition Seems very difficult, but if , it is equivalent to a 1-photon transition, with :  a frequency  =  L -  p  an effective Rabi frequency  eff =  p /  Two-level system : |g> |e> 00 LL vv |n> |n+1>      Lorentzian  n n+1 (  -  v ) 2 +  n n+1 2  eg (  L -  0 ) 2 +  eg 2

8. Raman transitions Position  √ I /  Compatible with  v Width :  <<  ’=  s  ’/2  ≈ 500 kHz  /2  ≈ 50 kHz Atomic observables not destroyed by spontaneous emission. Lamb-Dicke effect : Raman coherences survive.  n n+1 = (  n n +  n+1 n+1 )/2  n n = (2n+1)    ’ where   = 2 E R / h  v = 2  R /  v

9. Lamb Dicke Effect To evaluate the decay rate of the population of state |n> we have to consider the recoil due to spontaneous emission. The atom, close to R=0, absorbs a photon k L and emits a photon k sp The spatial part of the coupling is : exp i(k L -k sp )R We have to evaluate Assume k.R = k Z is small, and expand the exp exp i( k Z ) = 1 + i k Z + … Z = ( a + a † ) ( h / 2m  v ) 1/2 First order couples | n > only to | n+1 > and | n-1> Probability to go from | n > to | n+1 > : (n+1)  R /  v Probability to go from | n > to | n -1 > : n  R /  v Probability to leave | n > : (2n+1)  R /  v Average on k sp = 2 k L 2

10. Discussion The atom scatters a lot of photons. But the momentum of a photon is small compared to the width of the momentum distribution of the atomic state. The momentum distribution is not changed so much in a single event. The overlap of the modified distribution with the original one is large : 1 - (2n+1)  R /  v We are far in the Lamb-Dicke regime as :  R /2  = 2 kHz and  v /2  ≈ 100 kHz

11. Spectral analysis of the fluorescence Spontaneous Raman transitions Spontaneous red photon The temperature can be deduced from the ratio of the 2 side-bands. But one has to be careful, because of the optical thickness of the medium : the spontaneous photon acts as a probe for stimulated Raman transitions.

12. Recoil Induced Resonance Centered in  =0 Still narrower Strange shape Nothing to do with the lattice !

13. Raman trans. in momentum space E=p x 2 /2m pxpx Free atoms ; momentum kick :  p x = h k  Initial state : p x, E i =p x 2 /2m Final state : p x +  p x, E f =(p x +  p x ) 2 /2m Absorption :  [  (p x +  p x ) -  (p x )] (  - E f + E i ) 2 +  2 Assuming  p x «, and  small enough d  dp x p x = m  / k 

14. Classical picture Pump-probe interference pattern : very shallow potential moving at v x =  / k  Atoms slow down while climbing hills, and accelerate coming down. As the potential is very shallow, only atoms with a velocity close to v x =  / k  can feel the potential. If v x > 0, you have more atoms with v v x The density grating is following the interference pattern. For zero frequency components, the pump and the probe induce a density grating. The pump diffracts on that grating, and the diffracted wave interferes with the probegain or attenuation The signal for  is given by d  (the small param. is the potential depth). dp x p x = m  / k 

15. From 1D to 2D 1D : a pair of contra-propagating waves 2D : two pairs of contra-propagating waves Bad Idea ! Phase dependent potential 2 orthogonal standing-waves In phaseIn quadrature

16. Better idea Use just 3 waves with 120° Linear polarization out of plane Linear polarization in the plane m g =1/2 m g =-1/2

17. A few words about crystallography E tot =  E j exp-i k j.r = exp-i k 1.r  E j exp-i(k j - k 1 ).r For any translation R such that (k j - k 1 ).R = 2p j  the field is unchanged. R : vectors of the lattice (position space) {k j - k 1 } j>1 : basis of the reciprocal lattice (Brillouin zone). If they are (d-1) independent vectors. In the case of 2 orthogonal standing waves, (k 2 - k 1 )= (k 3 - k 1 ) + (k 4 - k 1 ) because k 2 = - k 1 and k 3 = - k 4 The problem of phase dependence is also related to that. With 3 beams in 2D, one can cancel the phases by an appropriate choice of the origins in space and in time.

18. 3D z E y E x

19. And then… 1D : 2 beams 2D : 3 beams 3D : 4 beams 4D : 5 beams… Where is the fourth dimension ? Consider a 3D restriction of a 4D periodic optical potential

20. 1D cut of a 2D potential 1 1 3 3 2 2 A 2D square lattice, but the atom can move only along a line. Depending on the slope of the line, one has different potentials.

21. Periodic, super-periodic & quasi-periodic potential 1 3 2 The slope is a simple rational number : Periodic potential The slope is a large integer : Super-periodic The slope is not a rational number : Quasi-periodic.

22. Lissajous r=f y /f x =1.5 r=25 r= √ 2

23. The angle  is small  ≈ 10 -2 rad Super-lattices

24. Fluorescence images With the extra beamWithout

25. Shadow image

26. Periodic, super-periodic & quasi-periodic potential 1 3 2 The slope is a simple rational number : Periodic potential The slope is a large integer : Super-periodic The slope is not a rational number : Quasi-periodic. In a quasi-periodic potential, the invariance by translation is lost. But a long range order remains.

27. Long range order FFT Similar patterns can be found in several places, but they differ slightly. Larger patterns larger distances U(x,y)=cos 2 x+cos 2 y y =  x V(x)=cos 2 x+cos 2 (  x) 2 frequencies {

28. Toy model for solid state physics Quasi-crystals with five-fold symmetry have been found in 1984. An alloy formed with Al, Pd and Mn, which are 3 metals (with a good conductivity), is almost an insulator (8 orders of magnitude). What is the role of the quasi-periodicity ? The conductivity is related to the mobility of the electrons in the potential of the ionic lattice. Ionic potentialOptical potential ElectronsAtoms Study the diffusion of atoms in a quasi-periodic potential ! }{}{

29. Optical lattice with 5-fold symmetry A 5-fold symmetry is incompatible with a translational invariance. i.e. you cannot cover the plane with pentagones. Penrose tilling.

30. It works ! One can measure :  the temperature  the life time  the vibration freq.  …

31. Spatial diffusion : method 1. Load the atoms from the MOT in the lattice 2. Wait  3. Take an image

32. Spatial diffusion :results Anisotropy in the diffusion by a factor of 2.

33. Far detuned lattices Red detuning : it works nicely ! but the atoms see a lot of light. Blue detuning : the atoms are in the dark ! for the same depth, less scattered photons Be careful in the design : the standard 4 beams configuration will not trap atoms. The total field is 0 along lines. 3D trap with two beams.

34. 1D array of ring-shaped traps. 2 contrapropagating beams with different transverse shapes, and blue detuning : r 0 I Hollow beam Gaussian beam r 0 : possible destructive interference r 0 U z U /2 r0r0

35. A conical lens r Intensité Expérience Simulation

36. The hollow beam CCD Lens Mask Telescope Fluorescence of the hot atoms with the hollow beam at resonance Ring diameter : 200 µm Ring width : 10 µm

37. The preliminary results Image of the atoms that remain in the lattice 80 ms after the end of the molasses.  = 2  20 GHz. Fraction (%) of the atoms that remain in the lattice vs time.

38. Instabilities in a MOT I       I 1 3 2 miroir cellule de césium MOT with retroreflected beams When the laser approches the resonance, some instabilities appear both on the shape and the position of the cloud. I will not consider here the instabilities and other rotating MOTs due to a misalignment of the beams

39. The shadow effect The beams are retro-reflected. The cloud of cold atoms absorbs part of the power. The backward beam is weaker than the forward one. The cloud is then pushed away from the center. We measure the displacement with a segmented photodiode. We can consider a 1D system with only global variables :  the number of atoms in the cloud, N  the motion of its center of mass, z and v. The repulsion due to multiple scattering has not to be taken into account, because it is an internal force. Assuming that the efficiency of the trapping process depends on the position of the center of mass, we obtain a set of three non-linear coupled equations. Numerical solutions.

40. The results Z N t (s) -0.8 -0.6 -0.4 -0.2 % pos. x 2000150010005000 ms 40x10 6 35 30 25 # atomes Theory Experiment